Economics 105: Statistics
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Transcript of Economics 105: Statistics
Economics 105: Statistics• Any questions?• Review #1 • GH 9 and GH 10 due on Wednesday
Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
• If the population standard deviation σ is unknown, we can substitute the sample standard deviation, sx
• This introduces extra uncertainty, since sx varies from sample to sample
• Use t distribution instead of the normal distribution
Confidence Interval for μ(σ Unknown)
Student’s t distribution• William Sealy Gosset was an Irish statistician who worked for Guinness Brewery in Dublin in the early 1900s. He was interested in the effects of various ingredients and temperature on beer, but only had a few batches of each “formula” to analyze. Thus, he needed a way to correctly treat SMALL SAMPLES in statistical analysis.
• Not supposed to be publishing, so used the pseudonym, “Student”
Student’s t distribution• The t is a family of distributions
• The shape depends on degrees of freedom (d.f.)– Number of observations that are free to vary after
sample mean has been calculated
d.f. = n - 1
Student’s t distribution
t0
t (df = 5)
t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Standard Normal
(t with df = ∞)
Note: t Z as n increases
• Assumptions– Population standard deviation is unknown– Population is normally distributed
• Use Student’s t distribution• Confidence Interval Estimate:
(where t is the critical value of the t distribution with n -1 degrees of freedom and an area of α/2 in each tail)
Confidence Interval for μ(σ Unknown)
(continued)
Student’s t Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0 2.920The body of the table contains t values, not probabilities
Let: n = 3 df = n - 1 = 2 = 0.10 /2 = 0.05
/2 = 0.05=TINV(0.1,2) 2.91998558
=TDIST(A3,2,1) 0.05=TDIST(2.92,2,1) 0.049999578
t distribution valuesWith comparison to the Z value
Confidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____
0.80 1.372 1.325 1.310 1.28
0.90 1.812 1.725 1.697 1.645
0.95 2.228 2.086 2.042 1.96
0.99 3.169 2.845 2.750 2.58
Note: t Z as n increases
Confidence Intervals for • A manufacturer produces bags of flour whose weights are normally distributed. A random sample of 25 bags was taken and their mean weight was 19.8 ounces with a sample standard deviation of 1.2 ounces. • Find and interpret a 99% confidence interval for the true average weight for all bags of flour produced by the company.
Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
Confidence Intervals for
• A random sample of 100 people shows
that 25 are left-handed.
• Form a 95% confidence interval for the
true proportion of left-handers
Confidence Intervals for • A random sample of 100 people shows
that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.
(continued)
Interpretation• We are 95% confident that the true percentage of
left-handers in the population is between 16.51% and 33.49%.
• Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed in repeated samples of size 100 in this manner are expected to contain the true proportion.
Confidence Intervals for • A random sample of 1000 people shows
that 250 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.
(continued)
Determining Sample Size
For the Mean
DeterminingSample Size
For theProportion
Sampling Error• The required sample size can be found to reach a
desired margin of error (e) with a specified level of confidence (1 - )
• The margin of error is also called sampling error– the amount of imprecision in the estimate of the population
parameter– the amount added and subtracted to the point estimate to
form the confidence interval
Determining Sample Size
For the Mean
DeterminingSample Size
Sampling error (margin of error)
Determining Sample Size
For the Mean
DeterminingSample Size
(continued)
Now solve for n to get
Determining Sample Size
• To determine the required sample size for the mean, you must know:
– The desired level of confidence (1 - ), which determines the critical Z value
– The acceptable sampling error, e
– The standard deviation, σ
(continued)
Required Sample Size Example
If = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?
(Always round up)
So the required sample size is n = 220
If σ is unknown
• If unknown, σ can be estimated when using the required sample size formula
–Use a value for σ that is expected to be at least as large as the true σ
–Select a pilot sample and estimate σ with the sample standard deviation, s
Determining Sample Size
DeterminingSample Size
For theProportion
Now solve for n to get
(continued)
Determining Sample Size• To determine the required sample size for
the proportion, you must know:– The desired level of confidence (1 - ), which
determines the critical Z value
– The acceptable sampling error, e
– The true proportion of “successes”, π
• π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.5)
(continued)
Required Sample Size Example
How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?
(Assume a pilot sample yields p = 0.12)
Required Sample Size Example
Solution:
For 95% confidence, use Z = 1.96
e = 0.03
p = 0.12, so use this to estimate π
So use n = 451
(continued)
What is a Hypothesis?• A hypothesis is a claim
(assumption) about a population parameter:
Example: The mean monthly cell phone bill of this city is μ = $42
Example: The proportion of adults in this city with cell phones is π = 0.68
The Null Hypothesis, H0
• States the claim or assertion to be tested
Example: The average number of TV sets in
U.S. Homes is equal to three ( )
• Is always about a population parameter, not about a sample statistic
The Null Hypothesis, H0
• Begin with the assumption that the null hypothesis is true–Similar to the notion of innocent until
proven guilty• Refers to the status quo• Always contains “=” , “≤” or “” sign• May or may not be rejected
(continued)
The Alternative Hypothesis, H1
• Is the opposite of the null hypothesis– e.g., The average number of TV sets in U.S.
homes is not equal to 3 ( H1: μ ≠ 3 )
• Challenges the status quo• Never contains the “=” , “≤” or “” sign• May or may not be proven• Is generally the hypothesis that the
researcher is trying to prove
Population
Claim: thepopulationmean age is 50.(Null Hypothesis:
REJECT
Supposethe samplemean age is 20: X = 20
SampleNull Hypothesis
20 likely if μ = 50?=Is
Hypothesis Testing Process
If not likely,
Now select a random sample
H0: μ = 50 )
X
Sampling Distribution of X
μ = 50If H0 is true
If it is unlikely that we would get a sample mean of this value ...
... then we reject the null
hypothesis that μ = 50.
Reason for Rejecting H0
20
... if in fact this were the population mean…
X